Research paper
Three-dimensional dynamics and synchronization of two coupled fluid-conveying pipes with intermediate springs

https://doi.org/10.1016/j.cnsns.2022.106777Get rights and content

Highlights

  • A three-dimensional dynamical model of coupled fluid-conveying pipes is developed.

  • Out-of-plane instability of the two parallel pipes occurs with increasing flow velocity.

  • The 3-D responses of the two pipes are quite different from the in-plane responses.

  • The two parallel pipes may experience synchronization in many cases.

Abstract

A system of two coupled oscillating structures may experience dynamical synchronization under coupling effects. Dynamical synchronization refers to the process in which two vibrating structures under coupling effects may display similar dynamics and vibrational behaviors even though their important parameters are not equivalent. In this study, the three-dimensional dynamics and synchronization behaviors of two coupled fluid-conveying pipes connected with linear springs are investigated. In the analytical model, the geometric nonlinearities due to axial elongation of the two pipes and the extension of linear springs are considered. The flow velocities of the internal fluid of the two pipes may be steady or pulsatile. In the case of the flow velocities of the two pipes being steady, the lowest several natural frequencies and the post-buckling behavior of the two-pipe system are obtained by changing the flow velocities in the two pipes. When one pipe conveys pulsatile fluid and the other conveys steady fluid, the dynamical bifurcation and synchronization behaviors of the two-pipe system are analyzed, with several typical synchronization patterns being explored by means of time traces, phase portraits, power spectral density (PSD) diagrams, Poincaré maps, etc. It is shown that making use of synchronization characteristics of the two-pipe system may provide a possible way to control the vibrations of the two-pipe system and to achieve some special engineering objectives.

Introduction

The phenomenon of synchronization was firstly observed by the Dutch physicist, Huygens, who, in 1665, stumbled upon the perfect synchronization of two side-by-side pendulums, opening a branch in mathematical science known as coupled-oscillator theory. Based on this theory, much attention has been paid to synchronization and its mechanisms, especially in living bodies, such as pacemaker cells in heart, neural networks that control human rhythms, simultaneous lighting and extinguishing of countless fireflies, and cricket singing in unison. Indeed, synchronization has been found to have very attractive applications and huge market potential value in the fields such as physical chemistry [1], [2], biology [3], mechanics [4], [5], [6], brain science [7], [8], [9], electronics, information science and secure communication [10], [11].

Dynamical synchronization of coupled oscillating systems is a typical branch of synchronizations in mechanics and hence has attracted great interest from many scholars. In 1982, based on the extended Lyapunov matrix method, Fujisaka et al. [12] investigated the stability criterion of synchronous vibration of coupled-oscillator systems. In their work, three coupled configurations were considered and the critical modes and state transitions were discussed. Gauthier et al. [13] analyzed a typical experimental system by the use of a commonly used theory for synchronization of coupled-oscillator systems and found that it could fail to estimate the accurate range of high-quality synchronization of the system. They also proposed a novel and feasible approach to predict the regime of high-quality synchronization. By the use of finite difference method, Barrón et al. [14] investigated the dynamical synchronization of a coupled self-excited system consisting of four elastic beams, and found that the coupled self-excited system was robust to dynamically and synchronously change the fluid–structure interaction properties.

It is known that chaos is an important phenomenon in nonlinear dynamics. Therefore, it is of great academic significance to study the synchronization behaviors of coupled chaotic oscillators. By using experimental and numerical methods, Taherion et al. [15] investigated how noise might influence the lag synchronization of two coupled chaotic oscillators. In their work, they found that when the noise in the coupled system is large, it is difficult to generate lag synchronization but a straightforward transition to complete synchronization would occur. Rosenblum et al. [16] investigated the dynamical synchronization transitions of two coupled self-excited oscillators. They found that two coupled self-excited oscillators may experience several transitions as the coupling strength increases. In addition to the study of lag synchronization, Boccaletti et al. [17] tried to introduce several main synchronization types, and analyzed the synchronization characteristics for different systems, including identical, nonidentical, structurally nonequivalent, chaotic and extended systems. Indeed, dynamical synchronization in coupled chaotic systems has been widely discussed in other literature [18], [19], [20], [21], [22], [23], [24], [25], [26], [27].

For the dynamical system of two or multiple coupled pipes, synchronization phenomenon in tube bundle in heat exchangers [28], [29] and multi-channel micro/nano tubes in MEMS was also observed. Because of coupling effects, the coupled fluid-conveying pipes may show wonderful dynamical synchronization behaviors. Ni et al. [30] investigated the dynamical synchronization of two coupled fluid-conveying pipes connected with linear springs. In their research, the effects of three typical fluid flow conditions and coupling stiffness on the dynamical behavior of the coupled two-pipe system were considered. It was found that spring stiffness has an important effect on the synchronization behaviors of the two-pipe system. Based on a similar physical model, Lü et al. [31] studied the dynamical synchronization and bifurcation behaviors of two fluid-conveying pipes connected by a nonlinear spring. In their work, the influences of spring location, spring stiffness and several important flow parameters on the bifurcation results were investigated. In addition, several synchronization patterns were detected in their work, such as lag, perfect, imperfect, strange and failed patterns. Combined with typical dynamic response results, including time histories, phase-plane plots, phase trajectories and PSD diagrams, these synchronization patterns were discussed in detail.

However, it should be pointed out that the vibrations of the two pipes considered in Refs. [30], [31] were assumed to be planar along the spring direction only. In fact, due to the action of the connecting springs, the stiffness of the two pipes along the spring direction is greater. Thus, the two pipes are more likely to vibrate perpendicular to the spring direction. With this consideration, it is necessary to visit the nonplanar vibration problem of the coupled two-pipe system. Moreover, to the best of the authors’ knowledge, the literature on this topic is very limited. Motived by this, the three-dimensional dynamics and synchronization behaviors of the dynamical system of two coupled fluid-conveying pipes connected by linear springs are explored in the present work, with three important issues being focused on: (i) Compared with a single fluid-conveying pipe, what is the difference between the coupled frequencies and modal shapes of the two coupled pipes and a single pipe? (ii) Under the action of the connecting springs, what is the difference between the dynamic responses of the two pipes along the spring direction and perpendicular to the spring direction? (iii) What synchronization and bifurcation behaviors can the two nonlinearly coupled pipes exhibit?

The remainder of this paper is organized as follows. In Section 2, a mathematical model for motions of the coupled two-pipe system is established, and then some key system parameters and initial conditions are given. In Section 3, the dynamic response of a single simply-supported pipe conveying steady fluid predicted in this study is compared with previous results, and good agreement is observed. The convergence analysis for the truncated mode number of Galerkin’s method is then conducted to determine a suitable truncated mode number. In Section 4, modal analysis for the coupled two-pipe system is carried out. By utilizing the linearized form of the governing equations, the natural frequency of the coupled pipe system is analyzed and the modal shapes of the lowest six modes are plotted. At the same time, based on the same physical model and system parameters, the model analysis is also conducted in the FEA (Finite Element Analysis) tool (ANSYS 2019 R3). The present theoretical results are compared with those calculated in the FEA software, and a good agreement is obtained. In Section 5, the nonlinear dynamics and dynamical synchronization of the two coupled fluid-conveying pipes are extensively studied. Finally, some important conclusions are drawn in Section 6.

Section snippets

Model description

The dynamical system of two fluid-conveying pipes under consideration is illustrated in Fig. 1. It consists of two identical pipes, denoted as pipe 1 and pipe 2, connected with three identical springs. The two pipes are of outer/inner diameter Do/Di, length L, flexural stiffness EI, and conveying steady or pulsatile fluid of velocity Ui1 and Ui2, respectively. The two pipes are simply supported at both ends and their initial centerlines are straight.

As for this two-pipe system, the following

Validation of a single pipe without intermediate spring

In this subsection, the validation of a single fluid-conveying pipe without intermediate springs will be checked by comparing the present result with that of Modarres-Sadeghi et al. [35]. In the work of Modarres-Sadeghi et al. [35], they investigated the post-buckling dynamics of a pinned–pinned pipe conveying steady fluid, with two coupled nonlinear equations for transverse and axial motions being considered.

In order to compare the present result with that of Modarres-Sadeghi et al. [35], a

Frequency analysis

In this section, a frequency analysis will be conducted for the coupled two-pipe system. After neglecting nonlinear terms and damping, one can obtain the linearized forms of the coupled nonlinear governing equations, as follows: 2v1τ2+2βu12v1ξτ+u122v1ξ2+4v1ξ4+ks2v1v2δξls=02v2τ2+2βu22v2ξτ+u222v2ξ2+4v2ξ4ksv12v2δξls=0 2w1τ2+2βu12w1ξτ+u122w1ξ2+4w1ξ4+ks2w1w2+l0r0w22w1δξls=02w2τ2+2βu22w2ξτ+u222w2ξ2+4w2ξ4ksw12w2+l0r02w2w1δξls=0

It can be seen from Eqs. 

Two pipes conveying steady fluid

In this subsection, the nonlinear dynamics of the two pipes conveying steady fluid are investigated. The complete nonlinear governing equations of the two-pipe system, i.e. Eqs. (30)–(33) are solved, with the dimensionless parameters and initial values defined in Eqs. (41), (42). In this case, the flow velocity of pipe 1 varies in the range of [0, 10], while the flow velocity of pipe 2 is fixed. The flow velocities of pipe 1 and pipe 2 are steady and are denoted as u1 and u2 respectively.

Fig. 8

Conclusions

In this paper, three-dimensional dynamics and dynamical synchronization behaviors of two fluid-conveying pipes coupled with linear springs are investigated. When deriving the governing equations of the coupled pipe system, the geometric nonlinearity due to centerline​ elongation of the two pipes and extension of the linear springs are considered. A four-mode truncation of the Galerkin’s method is employed to discrete the nonlinear governing equations. A fourth-order Runge–Kutta integration is

CRediT authorship contribution statement

T.L. Jiang: Investigation, Writing – original draft. L.B. Zhang: Writing – review & editing. Z.L. Guo: Writing – review & editing. H. Yan: Validation, Writing – review & editing. H.L. Dai: Supervision, Writing – review & editing. L. Wang: Supervision, Conceptualization, Writing – review & editing, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Nos. 12072119, 12102139 and 11972167).

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